Question

A tuning fork of \(512 \mathrm{~Hz}\) is used to produce resonance in a resonance tube experiment. The level of water at first resonance is $30.7 \mathrm{~cm}\( and at second resonance is \)63.2 \mathrm{~cm}$. What is the error in calculating velocity of sound? Assume the speed of sound $330 \mathrm{~m} / \mathrm{s}$. (A) \(58(\mathrm{~cm} / \mathrm{s})\) (B) \(204.1(\mathrm{~cm} / \mathrm{s})\) (C) \(280(\mathrm{~cm} / \mathrm{s})\) (D) \(110(\mathrm{~cm} / \mathrm{s})\)

Short answer

The error in calculating the velocity of sound is \(110(\mathrm{~cm} / \mathrm{s})\).

Step-by-step solution

Calculate the fundamental frequency of the tube

To calculate the fundamental frequency, we need to use the difference in water level at the first and second resonances, which is \( 63.2~cm - 30.7~cm = 32.5~cm \). We also need the frequency of the tuning fork, which is \(512Hz\). The relationship between the frequency of the tuning fork, the fundamental frequency, and the difference in water level is: \( n\Delta L = 2(2L_1 - \Delta L) \) Where \(n\) is the number of resonances (\(n = 2\) for the first resonance, and \(n = 3\) for the second resonance) and \(L_1\) is the water level at the first resonance. Since we only need to calculate the fundamental frequency, we'll use the data for the first resonance. For \(n=2\), the fundamental frequency (\(f_1\)) is given by: \( f_1 = \dfrac{f}{n-1} \) Where \( f \) is the frequency of the tuning fork.

Calculate the experimental speed of sound

Now that we have the fundamental frequency, we can calculate the experimental speed of sound using the water level at first resonance. Experimental speed of sound is given by: \(v = 2L_1 f_1 \)

Calculate the error in the experimental speed of sound

To find the error in the experimental speed of sound, we need to compare it with the given value, which is \(330 m/s\). The percentage error can be calculated as: \( \% \text{ error } = \dfrac{ \text{ Experimental speed of sound } - \text{ Actual speed of sound}}{ \text{ Actual speed of sound}} \times 100 \)

Convert the error to cm/s and find the answer

Now that we have the percentage error, we need to convert it to cm/s: \( \text{ Error } = ( \% \text{ error }\times \text{ Actual speed of sound}) / 100~cm/s \) Compare the error with the given options. The one closest to the calculated error is the answer.